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Many computer vision problems can be posed as learning a low-dimensional subspace from high dimensional data. The low rank matrix factorization (LRMF) represents a commonly utilized subspace learning strategy. Most of the current LRMF techniques are constructed on the optimization problem using L_1 norm and L_2 norm, which mainly deal with Laplacian and Gaussian noise, respectively. To make LRMF capable of adapting more complex noise, this paper proposes a new LRMF model by assuming noise as Mixture of Exponential Power (MoEP) distributions and proposes a penalized MoEP model by combining the penalized likelihood method with MoEP distributions. Such setting facilitates the learned LRMF model capable of automatically fitting the real noise through MoEP distributions. Each component in this mixture is adapted from a series of preliminary super- or sub-Gaussian candidates. An Expectation Maximization (EM) algorithm is also designed to infer the parameters involved in the proposed PMoEP model. The advantage of our method is demonstrated by extensive experiments on synthetic data, face modeling and hyperspectral image restoration.